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We consider the truncated multivariate normal distributions for which everycomponent is one-sided truncated. We show that this family of distributions isan exponential family. We identify $\mathcal{D}$, the corresponding naturalparameter space, and deduce that the family of distributions is not regular. Weprove that the gradient of the cumulant-generating function of the family ofdistributions remains bounded near certain boundary points in $\mathcal{D}$,and therefore the family also is not steep. We also consider maximum likelihoodestimation for $\boldsymbol{\mu}$, the location vector parameter, and$\boldsymbol{\Sigma}$, the positive definite (symmetric) matrix dispersionparameter, of a truncated non-singular multivariate normal distribution. Weprove that each solution to the score equations for$(\boldsymbol{\mu},\boldsymbol{\Sigma})$ satisfies the method-of-momentsequations, and we obtain a necessary condition for the existence of solutionsto the score equations.

Non-Steepness and Maximum Likelihood Estimation Properties of the Truncated Multivariate Normal Distributions